The problem of unbiased estimation, restricted only by the postulate of section 2, is considered here. For a chosen number $s > 1$, an unbiased estimate of a function $g$ on the parameter space, is said to be best at the parameter point $\theta_0$ if its $s$th absolute central moment at $\theta_0$ is finite and not greater than that for any other unbiased estimate. A necessary and sufficient condition is obtained for the existence of an unbiased estimate of $g$. When one exists, the best one is unique. A necessary and sufficient condition is given for the existence of only one unbiased estimate with finite $s$th absolute central moment. The $s$th absolute central moment at $\theta_0$ of the best unbiased estimate (if it exists) is given explicitly in terms of only the function $g$ and the probability densities. It is, to be more precise, specified as the l.u.b. of certain set $\mathcal{a}$ of numbers. The best estimate is then constructed (as a limit of a sequence of functions) with the use of only the data (relating to $g$ and the densities) associated with any particular sequence in $\mathcal{a}$ which converges to the l.u.b. of $\mathcal{a}$. The case $s = \infty$ is considered apart. The case $s = 2$ is studied in greater detail. Previous results of several authors are discussed in the light of the present theory. Generalizations of some of these results are deduced. Some examples are given to illustrate the applications of the theory.

A number of practical problems involve the solution of a mathematical problem of the class described in the classical language of probability theory as follows: "A number of balls are independently distributed among a number of boxes, how many boxes contain no balls, 1 ball, 2 balls, 3 balls, and so on." Problems arising in the oxidation of rubber and the genetics of bacteria are discussed as applications. A method is given of solving problems of this sort when "how many" is adequately answered by the calculation of means, variances, covariances, third moments, etc. The method is applied to a number of the simplest cases, where the number of balls is fixed, binomially distributed or Poisson and where the "sizes" of the boxes are equal or unequal.

The present paper is concerned with the power function of the classical tests associated with the normal distribution. Proofs of Hsu, Simaika, and Wald are simplified in a general manner applicable to other tests involving the normal distribution. The set theoretic structure of several tests is characterized. A simple proof of the stringency of the classical test of a linear hypothesis is given.

The method of mixtures, explained in Section 2, is applied to derive the distribution functions of a positive quadratic form in normal variates and of the ratio of two independent forms of this type.

An expression for the joint distribution of serial correlation coefficients, circularly defined, has been derived. It has been shown that this distribution possesses properties similar to those already encountered in the distribution of a single serial correlation coefficient, i.e. it is defined by different function forms for various subregions. The distribution thus found is of little use for computational purposes. Consequently, approximate forms have been investigated and the suitability of the ordinary partial correlation coefficient for large-sample testing has been inferred.

This paper deals with the following problem: Suppose a population of known size $N$ is subdivided into an unknown number of mutually exclusive classes. It is assumed that the class in which an element is contained may be determined, but that the classes are not ordered. Let us draw a random sample of $n$ elements without replacement from the population. The problem is to estimate the total number $K$ of classes which subdivide the population on the basis of the sample results and our knowledge of the population size. There is exactly one real valued statistic $S$ which is an unbiased estimate of $K$ when the sample size $n$ is not less than the maximum number $q$ of elements contained in any class. The restriction placed upon $q$ is unimportant for many practical problems where either there is a reasonably low bound for $q$ or those classes containing more than $n$ elements are known. An unbiased estimate does not exist when there is no such knowledge. Since the unbiased estimate can be very unreasonable, modifications of $S$ are considered. The statistic $T' = \big\{S' = N - \frac{N(n - 1)}{n(n - 1)} x_2,\quad\text{if} S' \geq \sum^n_{i=1} x_i,$ $\sum^n_{i=1} x_i,\quad\text{if} S' < \sum^n_{i=1} x_i,$ where $x_i$ is the number of classes containing $i$ elements in the sample, is the most suitable estimate, in comparison with three other statistics, for a hypothetical population. The case where each element in the population has an equal and independent chance of coming into the sample is used as a model for some sampling procedures and also as an approximation to the case of random sampling.

Let us consider a set of approximately random binary digits obtained by some experimental process. This paper outlines a method of compounding the digits of this set to obtain a smaller set of binary digits which is much more nearly random. The method presented has the property that the number of digits in the compounded set is a reasonably large fraction (say of the magnitude $\frac{1}{3}$ or $\frac{1}{4}$) of the original number of digits. If a set of very nearly random decimal digits is required, this can be obtained by first finding a set of very nearly random binary digits and then converting these digits to decimal digits. The concept of "maximum bias" is introduced to measure the degree of randomness of a set of digits. A small maximum bias shows that the set is very nearly random. The question of when a table of approximately random digits can be considered suitable for use as a random digit table is investigated. It is found that a table will be satisfactory for the usual types of situations to which a random digit table is applied if the reciprocal of the number of digits in the table is noticeably greater than the maximum bias of the table.